Complement Of Graph Research Articles

Doubly connected bi-domination in graphs

Let [Formula: see text] be a finite graph, simple, undirected and has no isolated vertex. A dominating subset [Formula: see text] of [Formula: see text] is said a bi-dominating set, if every vertex of it dominates two vertices of [Formula: see text]. The bi-domination number of [Formula: see text], denoted by [Formula: see text] is the minimum cardinality over all bi-dominating sets in [Formula: see text]. In this paper, a certain modified bi-domination parameter called doubly connected bi-domination and its inverse are introduced. Several bounds and properties are studied here. These modified dominations are applied and evaluated for several well-known graphs and complement graphs.

On the Complexity of Colouring Antiprismatic Graphs

A graph G is prismatic if for every triangle T of G, every vertex of G not in T has a unique neighbour in T. The complement of a prismatic graph is called \emph{antiprismatic}. The complexity of colouring antiprismatic graphs is still unknown. Equivalently, the complexity of the clique cover problem in prismatic graphs is not known. Chudnovsky and Seymour gave a full structural description of prismatic graphs. They showed that the class can be divided into two subclasses: the orientable prismatic graphs, and the non-orientable prismatic graphs. We give a polynomial time algorithm that solves the clique cover problem in every non-orientable prismatic graph. It relies on the the structural description and on later work of Javadi and Hajebi. We give a polynomial time algorithm which solves the vertex-disjoint triangles problem for every prismatic graph. It does not rely on the structural description.

The Q-generating Function for Graphs with Application

For a simple connected graph G, the Q-generating function of the numbers $$N_k$$ of semi-edge walks of length k in G is defined by $$W_Q(t)=\sum \nolimits _{k = 0}^\infty {N_k t^k }$$ . This paper reveals that the Q-generating function $$W_Q(t)$$ may be expressed in terms of the Q-polynomials of the graph G and its complement $$\overline{G}$$ . Using this result, we study some Q-spectral properties of graphs and compute the Q-polynomials for some graphs obtained from various graph operations, such as the complement graph of a regular graph, the join of two graphs and the (edge)corona of two graphs. As another application of the Q-generating function $$W_Q(t)$$ , we also give a combinatorial interpretation of the Q-coronal of G, which is defined to be the sum of the entries of the matrix $$(\lambda I_n-Q(G))^{-1}$$ . This result may be used to obtain the many alternative calculations of the Q-polynomials of the (edge)corona of two graphs. Further, we also compute the Q-generating functions of the join of two graphs and the complete multipartite graphs.

Exposed circuits, linear quotients, and chordal clutters

A graph G is said to be chordal if it has no induced cycles of length four or more. In a recent preprint Culbertson, Guralnik, and Stiller give a new characterization of chordal graphs in terms of sequences of what they call ‘edge-erasures’. We show that these moves are in fact equivalent to a linear quotient ordering on IG‾, the edge ideal of the complement graph. Known results imply that IG‾ has linear quotients if and only if G is chordal, and hence this recovers an algebraic proof of their characterization. We investigate higher-dimensional analogues of this result, and show that in fact linear quotients for more general circuit ideals of d-clutters can be characterized in terms of removing exposed circuits in the complement clutter. Restricting to properly exposed circuits can be characterized by a homological condition. This leads to a notion of higher dimensional chordal clutters which borrows from commutative algebra and simple homotopy theory. The interpretation of linear quotients in terms of shellability of simplicial complexes also has applications to a conjecture of Simon regarding the extendable shellability of k-skeleta of simplices. Other connections to combinatorial commutative algebra, chordal complexes, and hierarchical clustering algorithms are explored.

On the ordering of the Kirchhoff indices of the complements of trees and unicyclic graphs

The Kirchhoff index K f (G) of a graph G is defined to be the sum of the resistance distances between all pairs of vertices of G. In this paper, we develop a novel method for ordering the Kirchhoff indices of the complements of trees and unicyclic graphs. With this method, we determine the first five maximum values of K f ( $$\overline{T}$$ ) and the first four maximum values of K f (Ū), where $$\overline{T}$$ and Ū are the complements of a tree T and unicyclic graph U, respectively.

The Connectivity of a Bipartite Graph and Its Bipartite Complementary Graph

In 1956, Nordhaus and Gaddum gave lower and upper bounds on the sum and the product of the chromatic number of a graph and its complement, in terms of the order of the graph. Since then, any bound on the sum and/or the product of an invariant in a graph [Formula: see text] and the same invariant in the complement [Formula: see text] of [Formula: see text] is called a Nordhaus-Gaddum type inequality or relation. The Nordhaus-Gaddum type inequalities for connectivity have been studied by several authors. For a bipartite graph [Formula: see text] with bipartition ([Formula: see text]), its bipartite complementary graph [Formula: see text] is a bipartite graph with [Formula: see text] and [Formula: see text] and [Formula: see text]. In this paper, we obtain the Nordhaus-Gaddum type inequalities for connectivity of bipartite graphs and its bipartite complementary graphs. Furthermore, we prove that these inequalities are best possible.

Inverses of non-bipartite unicyclic graphs with a unique perfect matching

The class of non-bipartite unicyclic graphs with a unique perfect matching, denoted by , is considered in this article. This article describes the entries of the inverse of the adjacency matrix of a graph in . It is proved that the inverse graph of a graph in is always non-bipartite. In this article, we characterize the graphs in whose inverses are mixed graphs. Among all such graphs in we identify those graphs whose inverses are quasi-bipartite. Furthermore, characterizations of unicyclic graphs in possessing unicyclic and bicyclic inverses are also provided in this article.

Energy of inverse graphs of dihedral and symmetric groups

Let (G,∗) be a finite group and S={x∈G|x≠x−1} be a subset of G containing its non-self invertible elements. The inverse graph of G denoted by Γ(G) is a graph whose set of vertices coincides with G such that two distinct vertices x and y are adjacent if either x∗y∈S or y∗x∈S. In this paper, we study the energy of the dihedral and symmetric groups, we show that if G is a finite non-abelian group with exactly two non-self invertible elements, then the associated inverse graph Γ(G) is never a complete bipartite graph. Furthermore, we establish the isomorphism between the inverse graphs of a subgroup Dp of the dihedral group Dn of order 2p and subgroup Sk of the symmetric groups Sn of order k! such that 2p = n!~(p,n,k geq 3~text {and}~p,n,k in mathbb {Z}^{+}).

The First and Second Zagreb Index of Complement Graph and Its Applications of Molecular Graph

In this paper, some basic mathematical operation for the second Zagreb indices of graph containing the join and strong product of graph operation, and the rst and second Zagreb indices of complement graph operations such as cartesian product G1 G2, composition G1 G2, disjunction G1 _ G2, symmetric dierence G1 G2, join G1 + G2, tensor product G1 G2, and strong product G1 G2 will be explained. The results are applied to molecular graph of nanotorus and titania nanotubes.

Harary energy of complement of line graphs of regular graphs

The Harary matrix of a graph $G$ is defined as $H(G) = [h_{ij}]$, where $h_{ij} =\frac{1}{d(v_i, v_j)}$, if $i \neq j$ and $h_{ij} = 0$, otherwise, where $d(v_i, v_j)$ is the distance between the vertices $v_i$ and $v_j$ in $G$. The $H$-energy of $G$ is defined as the sum of the absolute values of the eigenvalues of Harary matrix. Two graphs are said to be $H$-equienergetic if they have same $H$-energy. In this paper we obtain the $H$-energy of the complement of line graphs of certian regular graphs interms of the order and regularity of a graph and thus constructs pairs of $H$-equienergetic graphs of same order and having different $H$-eigenvalues.

Some Results on the 3-Vertex-Rainbow Index of a Graph

Let G be a nontrivial connected graph with a vertex-coloring c: $$V(G)\rightarrow \{1,2,\ldots ,q\},q\in N$$ . For a set $$S\subseteq V(G)$$ and $$|S|\ge 2$$ , a subtree T of G satisfying $$S\subseteq V(T)$$ is said to be an S-Steiner tree or simply S-tree. The S-tree T is called a vertex-rainbow S-tree if the vertices of $$V(T)\setminus S$$ have distinct colors. Let k be a fixed integer with $$2\le k\le |V(G)|$$ , if every k-subset S of V(G) has a vertex-rainbow S-tree, then G is said to be vertex-rainbow k-tree connected. The k-vertex-rainbow index of G, denoted by $$rvx_{k}(G)$$ , is the minimum number of colors that are needed in order to make G vertex-rainbow k-tree connected. In this paper, we study the 3-vertex-rainbow index of unicyclic graphs and complementary graphs, respectively.

Construction for the Sequences of Q-Borderenergetic Graphs

This research intends to construct a signless Laplacian spectrum of the complement of any k-regular graph G with order n. Through application of the join of two arbitrary graphs, a new class of Q-borderenergetic graphs is determined with proof. As indicated in the research, with a regular Q-borderenergetic graph, sequences of regular Q-borderenergetic graphs can be constructed. The procedures for such a construction are determined and demonstrated. Significantly, all the possible regular Q-borderenergetic graphs of order 7<n≤10 are determined.

Captive domination in graphs

In this paper, a new definition of graph domination called “Captive Domination” is introduced. The proper subset of the vertices of a graph [Formula: see text] is a captive dominating set if it is a total dominating set and each vertex in this set dominates at least one vertex which doesn’t belong to the dominating set. The inverse captive domination is also introduced. The lower and upper bounds for the number of edges of the graph are presented by using the captive domination number. Moreover, the lower and upper bounds for the captive domination number are found by using the number of vertices. The condition when the total domination and captive domination number are equal to two is discussed and obtained results. The captive domination in complement graphs is discussed. Finally, the captive dominating set of paths and cycles are determined.

Line zero divisor graphs

Let [Formula: see text] be a commutative ring and [Formula: see text] be the zero divisor graph of [Formula: see text]. In this paper, we investigate when the zero divisor graph is a line graph. We completely present all commutative rings which their zero divisor graphs are line graphs. Also, we study when the zero divisor graph is the complement of a line graph.

Grundy coloring in some subclasses of bipartite graphs and their complements

A vertex v is a Grundy vertex with respect to a proper k-coloring c of a graph G=(V,E) if v has a neighbor of color j for every j(1≤j<i≤k), where i=c(v). A proper k-coloring c of G is called a Grundy k-coloring of G if every vertex is a Grundy vertex with respect to c and the largest integer k such that G admits a Grundy k-coloring is called the Grundy number of G which is denoted as Γ(G). Given a graph G and an integer k, the Grundy number decision problem is to decide whether Γ(G)≥k. The Grundy number decision problem is known to be NP-complete for bipartite graphs and complement of bipartite graphs. In this paper, we strengthen this result by showing that this problem remains NP-complete for perfect elimination bipartite graphs as well as for complement of perfect elimination bipartite graphs. Further, we give a linear-time algorithm to find the Grundy number of chain graphs, which is a proper subclass of the class of perfect elimination bipartite graphs. We also give a linear-time algorithm to find the Grundy number in complements of chain graphs. A partial Grundy coloring of a graph G is a proper k-coloring of G such that there is at least one Grundy vertex with each color i, 1≤i≤k and the partial Grundy number of G, ∂Γ(G), is the largest integer k such that G admits a partial Grundy k-coloring. Given a graph G and an integer k, the partial Grundy number decision problem is to decide whether ∂Γ(G)≥k. It is known that the partial Grundy number decision problem is NP-complete for bipartite graphs. In this paper, we prove that this problem is NP-complete in the complements of bipartite graphs by showing that the Grundy number and partial Grundy number are equal in complements of bipartite graphs.

Regularity, Rees algebra, and Betti numbers of certain cover ideals

Let $$S={\textsf {k}}[X_1,\dots , X_n]$$ be a polynomial ring, where $${\textsf {k}}$$ is a field. This article deals with the defining ideal of the Rees algebra of a squarefree monomial ideal generated in degree $$n-2$$ . As a consequence, we prove that Betti numbers of powers of the cover ideal of the complement graph of a tree do not depend on the choice of the tree. Further, we study the regularity and Betti numbers of powers of cover ideals associated to certain graphs.

Detecting disease-related SNP loci based on GSP

A large number of studies have shown that susceptibility to some diseases may be related to some SNP (Single Nucleotide Polymorphism) loci. Accurate location of disease-related SNP loci can help people understand the pathogenesis of diseases and prevent them from happening. Based on Graph Signal Processing (GSP) theory, this paper proposes a novel method called GSP to detect disease-related SNP loci. GSP method is divided into five steps. The first step is to establish a graph signal model of all SNP loci. The second step is to extract the high-frequency components of graph signal with a high-pass filter. The third step is to transform the filtered graph signal with graph Fourier transformation. The fourth step is to extract the graph signal corresponding to high frequency with inverse graph Fourier transformation. The fifth step is to transform the extracted signal with Gaussian distribution and use Pauta criterion to screen out disease-related SNP loci. The experimental results show that GSP method can detect all disease-related SNP loci of simulation data and Genetic Disease A (GDA) and three of the four disease-related SNP loci of Age-related Macular Degeneration (AMD) with excellent performance. GSP method provides a new idea for the identification of pathogenicity loci.

Hausdorff property of G^++−, G^+−+ and their complement graphs

A simple graph G is said to be Hausdorff graph if for any two vertices u and v of G satisfy at least one of the following conditions: [1] both u and v are isolated [2] either u or v is isolated[3] there exists two non-adjacent edges e1 and e2 of G such that e1 is incident with u and e2is incident with v. In this paper, we discuss Hausdorff property on some specific transformationgraphs namely G++-, G+-+, G--+ and G-+- .

DTiGEMS+: drug\u2013target interaction prediction using graph embedding, graph mining, and similarity-based techniques

In silico prediction of drug–target interactions is a critical phase in the sustainable drug development process, especially when the research focus is to capitalize on the repositioning of existing drugs. However, developing such computational methods is not an easy task, but is much needed, as current methods that predict potential drug–target interactions suffer from high false-positive rates. Here we introduce DTiGEMS+, a computational method that predicts Drug–Target interactions using Graph Embedding, graph Mining, and Similarity-based techniques. DTiGEMS+ combines similarity-based as well as feature-based approaches, and models the identification of novel drug–target interactions as a link prediction problem in a heterogeneous network. DTiGEMS+ constructs the heterogeneous network by augmenting the known drug–target interactions graph with two other complementary graphs namely: drug–drug similarity, target–target similarity. DTiGEMS+ combines different computational techniques to provide the final drug target prediction, these techniques include graph embeddings, graph mining, and machine learning. DTiGEMS+ integrates multiple drug–drug similarities and target–target similarities into the final heterogeneous graph construction after applying a similarity selection procedure as well as a similarity fusion algorithm. Using four benchmark datasets, we show DTiGEMS+ substantially improves prediction performance compared to other state-of-the-art in silico methods developed to predict of drug-target interactions by achieving the highest average AUPR across all datasets (0.92), which reduces the error rate by 33.3% relative to the second-best performing model in the state-of-the-art methods comparison.

Unified Spectral Hamiltonian Results of Balanced Bipartite Graphs and Complementary Graphs

There have been researches on sufficient spectral conditions for Hamiltonian properties and path-coverable properties of graphs. Utilizing the Bondy–Chvatal closure, we provide a unified approach to study sufficient graph eigenvalue conditions for these properties and sharpen several former spectral Hamiltonian results on balanced bipartite graphs and complementary graphs.